Non-linear models have long been used for the purpose of predicting the operation of a plant or system. Empirical nonlinear models are trained on a historical data set that is collected over long periods of time during the operation of the plant or system. This historical data set only covers a limited portion of the “input space” of the plant, i.e., only that associated with data that is collected within the operating range of the plant when the data was sampled. The plant or system typically has a set of manipulatable variables (MV) that can be controlled by the operator. There are also some variables that are referred to as “disturbance variables” (DV) which are variables that cannot be changed by the operator, such as temperature, humidity, fuel composition, etc. Although these DVs have an affect on the operation of the plant or system, they cannot be controlled and some can not be measured. The plant will typically have one or more outputs that are a function of the various inputs, the manipulatable variables and the disturbance variables. These outputs are sometimes referred to as “controlled variables” (CVs). If the historical data set constitutes a collection of the measured output values of the plant/system and the associated input values thereto, a non-linear network can be trained on this data set to provide a training model that can predict the output based upon the inputs. One type of such non-linear model is a neural network that is comprised of an input layer and an output layer with a hidden layer disposed there between, which hidden layer stores a representation of the plant or system, wherein the input layer is mapped to the output layer through the stored representation in the hidden layer. Another is a support vector machine (SVM). These are conventional networks. However, since the unmeasurable disturbances cannot be measured, the model can not be trained on these variables as a discrete input thereto; rather, the data itself embodies these errors.
One application of a non-linear model is that associated with a control system for controlling a power plant to optimize the operation thereof in view of desired output variables. One output variable that is of major concern in present day power plants is nitrogen oxide (NOx). Nitrogen oxides lead to ozone formation in the lower atmosphere which is a measurable component of urban photochemical smog, acid rain, and human health problems such as lung damage and people with lung disease. NOx levels are subject to strict guidelines for the NOx emissions throughout the various industrialized areas of the country. For example, the current U.S. administration has proposed legislation to reduce NOx emissions from power plants by 50% over the next decade.
These Nitrogen oxides are formed during combustion and emitted into the atmosphere as a component of the exit flue gas. The level of NOx formation is dependent upon the local fuel/air stochiometric ratio in the furnace of a coal fired plant, as well as the amount of nitrogen in the fuel. A fuel rich environment (high fuel/air stochiometric ratio) leads to low NOx formation but high carbon monoxide (CO) amounts, another regulated gas. Therefore, an optimal fuel/air ratio needs to be determined to minimize NOx while maintaining CO below a specified constraint. In a coal fired plant, both fuel and air are input into the furnace at multiple points. Furthermore, a non-linear relationship between fuel and air inputs and NOx formation has been observed, such that optimization of the NOx levels, while observing the CO constraint, requires a non-linear, multiple input, multiple output (MIMO) controller. These optimizers typically utilize a nonlinear neural network model, such as one that models NOx formation, such that prediction of future set points for the purpose of optimizing the operation of the plant or system is facilitated given a cost function, constraints, etc.
As noted herein above, to develop accurate models for NOx, it is necessary to collect actual operating data from the power plant over the feasible range of operation. Since plants typically operate in a rather narrow range, historical data does not provide a rich data set over a large portion of the input space of the power plant for developing empirical models. To obtain a valid data set, it is necessary to perform a series of tests at the plan. Each test involves initially moving approximately five to ten inputs of the furnace to a designated set point. This set point is then held for 30–60 minutes until the process has obtained a stable steady state value. This is due to the fact that the desired nonlinear model is a steady state model. At that time, the NOx and input set points are recorded, this forming a single input-output pattern in the historical data set. It may take two engineers three to four weeks of round-the-clock testing to collect roughly 200–400 data points needed to build a model.
The training of the model utilizes a standard feed-forward neural network using a steepest descent approach. Since only data from steady state data points are used, the model is an approximation of the steady state mapping from the input to the output. Utilizing a standard back propagation approach, the accuracy of the model is affected by the level of the unmeasured disturbance inherent to any process. Because the major source of nitrogen for the formation of NOx in a combustion process at coal fired plants is the fuel, variations in the amount of nitrogen in the coal can cause slow variations in the amount of NOx emissions. Furthermore, most power plants do not have on-line coal analysis instruments needed to measure the nitrogen content of the coal. Thus, the variation of nitrogen content along with other constituents introduces a significant, long-term, unmeasured disturbance into the process that affects the formation of NOx.